3.3026 \(\int \frac{1}{\left (a+b \left (c x^n\right )^{2/n}\right )^3} \, dx\)

Optimal. Leaf size=98 \[ \frac{3 x \left (c x^n\right )^{-1/n} \tan ^{-1}\left (\frac{\sqrt{b} \left (c x^n\right )^{\frac{1}{n}}}{\sqrt{a}}\right )}{8 a^{5/2} \sqrt{b}}+\frac{3 x}{8 a^2 \left (a+b \left (c x^n\right )^{2/n}\right )}+\frac{x}{4 a \left (a+b \left (c x^n\right )^{2/n}\right )^2} \]

[Out]

x/(4*a*(a + b*(c*x^n)^(2/n))^2) + (3*x)/(8*a^2*(a + b*(c*x^n)^(2/n))) + (3*x*Arc
Tan[(Sqrt[b]*(c*x^n)^n^(-1))/Sqrt[a]])/(8*a^(5/2)*Sqrt[b]*(c*x^n)^n^(-1))

_______________________________________________________________________________________

Rubi [A]  time = 0.0709716, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{3 x \left (c x^n\right )^{-1/n} \tan ^{-1}\left (\frac{\sqrt{b} \left (c x^n\right )^{\frac{1}{n}}}{\sqrt{a}}\right )}{8 a^{5/2} \sqrt{b}}+\frac{3 x}{8 a^2 \left (a+b \left (c x^n\right )^{2/n}\right )}+\frac{x}{4 a \left (a+b \left (c x^n\right )^{2/n}\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*(c*x^n)^(2/n))^(-3),x]

[Out]

x/(4*a*(a + b*(c*x^n)^(2/n))^2) + (3*x)/(8*a^2*(a + b*(c*x^n)^(2/n))) + (3*x*Arc
Tan[(Sqrt[b]*(c*x^n)^n^(-1))/Sqrt[a]])/(8*a^(5/2)*Sqrt[b]*(c*x^n)^n^(-1))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 8.313, size = 82, normalized size = 0.84 \[ \frac{x}{4 a \left (a + b \left (c x^{n}\right )^{\frac{2}{n}}\right )^{2}} + \frac{3 x}{8 a^{2} \left (a + b \left (c x^{n}\right )^{\frac{2}{n}}\right )} + \frac{3 x \left (c x^{n}\right )^{- \frac{1}{n}} \operatorname{atan}{\left (\frac{\sqrt{b} \left (c x^{n}\right )^{\frac{1}{n}}}{\sqrt{a}} \right )}}{8 a^{\frac{5}{2}} \sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b*(c*x**n)**(2/n))**3,x)

[Out]

x/(4*a*(a + b*(c*x**n)**(2/n))**2) + 3*x/(8*a**2*(a + b*(c*x**n)**(2/n))) + 3*x*
(c*x**n)**(-1/n)*atan(sqrt(b)*(c*x**n)**(1/n)/sqrt(a))/(8*a**(5/2)*sqrt(b))

_______________________________________________________________________________________

Mathematica [A]  time = 4.40314, size = 0, normalized size = 0. \[ \int \frac{1}{\left (a+b \left (c x^n\right )^{2/n}\right )^3} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[(a + b*(c*x^n)^(2/n))^(-3),x]

[Out]

Integrate[(a + b*(c*x^n)^(2/n))^(-3), x]

_______________________________________________________________________________________

Maple [F]  time = 0.044, size = 0, normalized size = 0. \[ \int \left ( a+b \left ( c{x}^{n} \right ) ^{2\,{n}^{-1}} \right ) ^{-3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b*(c*x^n)^(2/n))^3,x)

[Out]

int(1/(a+b*(c*x^n)^(2/n))^3,x)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((c*x^n)^(2/n)*b + a)^(-3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.265094, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (b^{2} c^{\frac{4}{n}} x^{4} + 2 \, a b c^{\frac{2}{n}} x^{2} + a^{2}\right )} \log \left (\frac{2 \, a b c^{\frac{2}{n}} x +{\left (b c^{\frac{2}{n}} x^{2} - a\right )} \sqrt{-a b c^{\frac{2}{n}}}}{b c^{\frac{2}{n}} x^{2} + a}\right ) + 2 \,{\left (3 \, b c^{\frac{2}{n}} x^{3} + 5 \, a x\right )} \sqrt{-a b c^{\frac{2}{n}}}}{16 \,{\left (a^{2} b^{2} c^{\frac{4}{n}} x^{4} + 2 \, a^{3} b c^{\frac{2}{n}} x^{2} + a^{4}\right )} \sqrt{-a b c^{\frac{2}{n}}}}, \frac{3 \,{\left (b^{2} c^{\frac{4}{n}} x^{4} + 2 \, a b c^{\frac{2}{n}} x^{2} + a^{2}\right )} \arctan \left (\frac{\sqrt{a b c^{\frac{2}{n}}} x}{a}\right ) +{\left (3 \, b c^{\frac{2}{n}} x^{3} + 5 \, a x\right )} \sqrt{a b c^{\frac{2}{n}}}}{8 \,{\left (a^{2} b^{2} c^{\frac{4}{n}} x^{4} + 2 \, a^{3} b c^{\frac{2}{n}} x^{2} + a^{4}\right )} \sqrt{a b c^{\frac{2}{n}}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((c*x^n)^(2/n)*b + a)^(-3),x, algorithm="fricas")

[Out]

[1/16*(3*(b^2*c^(4/n)*x^4 + 2*a*b*c^(2/n)*x^2 + a^2)*log((2*a*b*c^(2/n)*x + (b*c
^(2/n)*x^2 - a)*sqrt(-a*b*c^(2/n)))/(b*c^(2/n)*x^2 + a)) + 2*(3*b*c^(2/n)*x^3 +
5*a*x)*sqrt(-a*b*c^(2/n)))/((a^2*b^2*c^(4/n)*x^4 + 2*a^3*b*c^(2/n)*x^2 + a^4)*sq
rt(-a*b*c^(2/n))), 1/8*(3*(b^2*c^(4/n)*x^4 + 2*a*b*c^(2/n)*x^2 + a^2)*arctan(sqr
t(a*b*c^(2/n))*x/a) + (3*b*c^(2/n)*x^3 + 5*a*x)*sqrt(a*b*c^(2/n)))/((a^2*b^2*c^(
4/n)*x^4 + 2*a^3*b*c^(2/n)*x^2 + a^4)*sqrt(a*b*c^(2/n)))]

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + b \left (c x^{n}\right )^{\frac{2}{n}}\right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b*(c*x**n)**(2/n))**3,x)

[Out]

Integral((a + b*(c*x**n)**(2/n))**(-3), x)

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (\left (c x^{n}\right )^{\frac{2}{n}} b + a\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((c*x^n)^(2/n)*b + a)^(-3),x, algorithm="giac")

[Out]

integrate(((c*x^n)^(2/n)*b + a)^(-3), x)